Diffusive Shock Acceleration and magnetic fields
in SNR
Tony Bell
University of Oxford Rutherford Appleton Laboratory
SN1006: A supernova remnant 7,000 light years from Earth X-ray (blue): NASA/CXC/Rutgers/G.Cassam-Chenai, J.Hughes et al; Radio (red): NRAO/AUI/GBT/VLA/Dyer, Maddalena & Cornwell; Optical (yellow/orange): Middlebury College/F.Winkler. NOAO/AURA/NSF/CTIO Schmidt & DSS
Mainly protons
Cosmic ray spectrum arriving at earth
(I) Galactic Supernova remnants
(III) Extra- galactic ?
(II) Probably galactic
knee ankle
CR populations
~E-2.6
~E-3
Why shocks?
Historical shell supernova remnants
Kepler 1604AD Tycho 1572AD
SN1006 Cas A 1680AD
Chandra observations
NASA/CXC/NCSU/ S.Reynolds et al.
NASA/CXC/Rutgers/ J.Warren & J.Hughes et al.
NASA/CXC/MIT/UMass Amherst/ M.D.Stage et al.
NASA/CXC/Rutgers/ J.Hughes et al.
Cassiopeia A
Radio (VLA)
Infrared (Spitzer)
Optical (Hubble)
X-ray (Chandra)
Mixture of line radiation & synchrotron continuum
Synchrotron in magnetic field ~ 0.1-1mG Radio (hν~10-5eV): electron energy ~1 GeV X-ray (hν~103eV): electron energy ~ 10 TeV
NASA/JPL-Caltech/O. Krause (Steward Observatory) NASA/JPL-Caltech/O. Krause (Steward Observatory) NASA/JPL NASA/JPL
NASA/JPL-Caltech/ O Krause(Steward Obs)
NASA/CXC/MIT/UMass Amherst/ M.D.Stage et al.
NASA/ESA/ Hubble Heritage (STScI/AURA))
chandra.harvard.edu/photo/0237/0237_radio.jpg
chandra.harvard.edu/photo/ 0237/0237_radio.jpg
SNR RX J1713.7-3946
HESS: γ-rays directly produced by TeV particles
Aharonian et al Nature (2004)
optical Radio jets
Centaurus A is the closest powerful radio galaxy (5Mpc)
Active galaxies
Cygnus A
X-ray (Chandra)
Strong shock: high Mach number
density
velocity
pressure
Upstream Downstream
shockuu = shockuu41
=0ρρ = 04ρρ =
0=kT 0=P 2
43
shockuP ρ=2
1163
shockpumZ
AkT
+
=
In shock rest frame
Conserved across shock (Rankine Hugoniot relations)
uρMass flux Momentum flux Energy flux
2uP ρ+
3
21
25 uPu ρ+
Shock turns kinetic streaming energy Into random thermal energy
Divert part of thermal energy Into high energy particles
Cosmic ray acceleration by shocks
Cosmic ray acceleration
High velocity plasma
Low velocity plasma
B2
B1
CR track
Due to scattering, CR recrosses shock many times Gains energy at each crossing
Shock acceleration energy spectrum: energy gain
Average fractional energy gained at each crossing is cp
pp sv21 =>∆<+>∆<
=∆εε
Change in momentum from upstream to downstream Mean increase in momentum Similar increase in momentum on recrossing into upstream
θ
shock
CR
ϑcosv1 c
pppp ∆=−′=∆
( )pcd
dpp s 2/v
sincos
sincos2/
0
2/
0 11 =
∆=>∆<
∫∫
π
π
ϑϑϑ
ϑϑϑ
Change in fluid velocity across shock
4v3
4v
vv sss =−=∆
( )pcp s 2/v2 =>∆<
Shock acceleration energy spectrum: loss rate
High velocity plasma
Low velocity plasma
Upstream ISM Downstream shocked plasma
B2
B1 B2>B1 Shock velocity: vs
CR density at shock: n
CR cross from upstream to downstream at rate nc/4 CR carried away downstream at rate nvdownstream = nvs/4 Mean number of shock crossings = (nc/4)/(nvs/4)=c/vs Fraction lost at each shock crossing is vs/c cn
n sv−=
∆
Fractional CR loss per shock crossing Fractional energy gain per shock crossing
Shock acceleration energy spectrum
cnn sv
−=∆
csv
=∆εε
High velocity plasma
Low velocity plasma
Upstream ISM Downstream shocked plasma
B2
B1 B2>B1 Shock velocity: vs
CR density at shock: n
Turn into differential equation εεεnn
ddn
−=∆∆
≈1−∝ εn
integrated spectrum
Differential energy spectrum εεεε ddN 2)( −∝
Derivation from Boltzmann equation
Krimskii 1977 Axford, Leer & Skadron 1977 Blandford & Ostriker 1978
The Vlasov-Fokker-Planck (VFP) equation
( ) )(.. fCfeftf
=∂∂
×+−∂∂
+∂∂
pBvE
rv
Vlasov equation (collisioness)
Collisions Fokker-Planck
1) Advection: at velocity v in r-space at velocity e(E+vxB) in p-space
2) Collisions: small angle scattering
zyxzyx dpdpdpdzdydxtpppzyxf ),,,,,,(= number of CR in phase space volume zyx dpdpdpdzdydx
VFP equation:
B on scale > CR Larmor radius
Due to B on scale < CR Larmor radius
Cosmic ray acceleration
B2
B1
CR track
Large scale B
Small scale B scatters CR
( ) )(.. fCfeftf
=∂∂
×+−∂∂
+∂∂
pBvE
rv
Parallel shock
CR track
)(. fCftf
=∂∂
+∂∂
rv
Only diffusion along B matters Large scale field irrelevant Same is if no large scale field
( ) )(.. fCfeftf
=∂∂
×+−∂∂
+∂∂
pBvE
rv
Redefine f in local fluid rest frame
CR track
)()v( fCpfp
xu
xfu
tf
xxx =
∂∂
∂∂
−∂∂
++∂∂
f fluid rest frame Fluid moves at velocity u
Frame transformation advection with fluid
u2=ushock/4 u1=ushock
pppfpff x)()( 10 +=To first approximation in u/c
isotropic drift
VFP equation reduces to
031
3v 0100 =
∂∂
∂∂
−∂∂
+∂∂
+∂∂
pfp
xu
xf
xfu
tf
10v fxf ν−=
∂∂
Scattering frequency
031
3v 0000 =
∂∂
∂∂
−
∂∂
∂∂
−∂∂
+∂∂
pfp
xu
xf
xxfu
tf ν
Advection Diffusion adiabatic compression
Sub-relativistic shocks: small u/c
downstreamupstream pfp
cuf
pfp
cuf
∂∂
−=
∂∂
− 021
011
u2=ushock/4 u1=ushock
pppfpff x)()( 10 +=
Downstream: no drift relative to background No escape upstream:
01 =f03 101 =+ fcfu
Boundary condition at shock
( ) 010
21 3 fupfpuu −=
∂∂
− 40
−∝ pf
Steady state solution
Acceleration efficiency
Efficiency
• Has to be efficient (10-50%) to explain galactic CR energy density
• Solar wind shocks can be >10% efficient
• Shock processes produce many suprathermal protons
At high efficiency: non-linear feedback onto shock
Drury & Voelk (1981)
CR pressure decelerates flow into shock
High efficiency: concave spectrum
Steep spectrum
Flat spectrum
SN1006 (Allen et al 2008)
Maximum CR energy
CR upstream of shock
02
2
=∂
∂−
∂∂
−=∂
∂xnD
xnu
tn crcrcr
Duxcr enn /
0=
shock
upstream
ncr
Balance between: • flow into shock • diffusion away from shock
Exponential density
Scaleheight
u
L=D/u
uDL =
Number of CR upstream: Rate CR cross shock: Average time spent upstream: (neglect time spent downstream) Average energy gain per shock crossing: Acceleration rate:
CR acceleration time
shocku
Dn upstream
cr cncr41
shock
upstream
cuD
t4
=∆
shock
upstream
ncr
u
L=D/u
22 )4/(44
shock
downstream
shock
upstream
uD
uD
+=τTime needed for acceleration (Lagage & Cesarsky)
cushock=
∆εε
upstream
shock
Du
t
2
4εε
=∆∆
downstream time
Acceleration time Diffusion coefficient
Maximum CR energy
2
8
shockuD
=τ
3cD λ
=
Rc
ushock
83
≤λ where τshockuR =
SNR radius
Smallest possible mfp: eBpcr=λ
Limit on CR momentum: eBRc
up shockcr 8
3=
Typically for young SNR ISM mag field: few µG ushock=c/30 R ~ 1017m
Max CR energy ~ 1014eV
under favourable assumptions
Limit on CR energy in eV: RBue
cpshock
cr
83
=
Hillas diagram
Get original version
(condition on RuB)
(Hillas, 1984)
Perpendicular shocks (Jokipii 1982, 1987)
CR trajectory at perpendicular shock (no scattering)
B into screen
shock
BuE shock ×−=
CR drift velocity
2vB
BEdrift
×=
CR gain energy by drifting in E field
CR acceleration at perpendicular shock
shock
CR trajectory divides into • Motion of gyrocentre
• Gyration about gyrocentre
Without diffusion: Every CR gets small adiabatic gain due to compression at shock
CR acceleration at perpendicular shock: with scattering
Strong scattering
Weak scattering
No scattering
Diffusive shock theory applies Provided gyrocentre diffuses over distances greater than Larmor radius during shock transit Same power law (see later)
Not to scale
CR acceleration at perpendicular shock
B
BuE shock ×−=
Transit between pole & equator: energy gain ~ BReueER shock=
Hillas parameter as with parallel shock: similar max CR energy
SN1006
The case of SN1006
B?
Polar x-ray synchrotron emission?
At perpendicular shocks • Acceleration is faster – potentially higher CR energy
• CR energy limited to euBR (Hillas) by space rather than time
• Injection is more difficult at a perpendicular shock
• CR scattering frequency has to be in right range Room for discussion!
(Rothenflug et al 2004)
CR scattering
what is the mean free path?
CR drive a ‘resonant’ instability
B
CR trajectory
Spatial resonance between wavelength and CR Larmor radius wave deflects CR CR current drives wave
Alfven wave
Skilling (1975)
xn
Upc
xIu
tI cr
M
A
∂∂
=∂∂
+∂∂ v
Wave growth (energy density I)
CR scattering
∂∂
∂∂
=∂
∂+
∂∂
xnD
xxnu
tn crcrcr
Icr
D g
π34
=
2
2
BBI δ
=
Turbulence upstream of shock
2shock
crAshock u
UMIρ
=Irg
π4mfp =
ncr
L=D/u Skilling (1975)
xn
Upc
xIu
tI cr
M
A
∂∂
=∂∂
+∂∂ v
Wave growth (amplitude I) CR scattering
∂∂
∂∂
=∂
∂+
∂∂
xnD
xxnu
tn crcrcr
Icr
D g
π34
=
Solution Ir
ucL g
shock34
=
Alfven Mach number ~1000
CR efficiency ~0.1 ?Implies?: mfp < Larmor radius Waves non-linear: I >> 1
Question: What does I > 1 tell us?
Kepler 1604AD Tycho 1572AD
SN1006 Cas A 1680AD 2
shock
crAshock u
UMIρ
=
Alfven Mach number ~1000 CR efficiency ~0.1
Evidence that magnetic field exceeds typical interstellar value
1) x-ray observations
2) Needed ti accelerate galactic CR to a few PeV
3) Theory breaks down because of large wave growth
Kepler 1604AD Tycho 1572AD
SN1006 Cas A 1680AD 2
shock
crAshock u
UMIρ
=
Alfven Mach number ~1000 CR efficiency ~0.1
1) x-ray observations
2) Needed ti accelerate galactic CR to a few PeV
3) Theory breaks down because of large wave growth
Need to look more closely at CR interaction with magnetic field
Evidence that magnetic field exceeds typical interstellar value
Diffusive Shock Acceleration and magnetic fields
in SNR
Tony Bell
University of Oxford Rutherford Appleton Laboratory
SN1006: A supernova remnant 7,000 light years from Earth X-ray (blue): NASA/CXC/Rutgers/G.Cassam-Chenai, J.Hughes et al; Radio (red): NRAO/AUI/GBT/VLA/Dyer, Maddalena & Cornwell; Optical (yellow/orange): Middlebury College/F.Winkler. NOAO/AURA/NSF/CTIO Schmidt & DSS
magnetic field amplification and
cosmic ray scattering
Shock
downstream upstream
CR streaming ahead of shock Excite instabilities Amplify magnetic field
CR pre-cursor
Streaming CR excite instabilities
SNR
CR
Streaming instabilities amplify magnetic field Lucek & Bell (2000)
CR treated as particles Thermal plasma as MHD
Electric currents carried by CR and thermal plasma
Density of 1015eV CR: ~10-12 cm-3
Current density: jcr ~ 10-18 Amp m-2
L R
CR pre-cursor jcr
CR current must be balanced by current carried by thermal plasma
jthermal = - jcr
jthermalxB force acts on plasma to balance jcrxB force on CR
Three equations control the instability
Equation for jcr in terms of perturbed B
1) 2) 3)
B CR
BjBjBBpdtdu
crcr ×−×−×∇×−−∇= ⊥ ||0
)(1µ
ρ
jxB driving force splits into two parts:
⊥crjcrj
BjBjBBptu
crcr ×−×−×∇×−−∇=∂∂
⊥ ||0
)(1µ
ρ
Resonant Alfven instability
B CR
⊥crjcrj
Bjcr ×⊥ drives Alfven waves
Perturbed cosmic ray current
BjBjBBptu
crcr ×−×−×∇×−−∇=∂∂
⊥ ||0
)(1µ
ρ
Non-resonant instability
Bjcr ×||
B CR
⊥crjcrj
dominates for shock acceleration in SNR
Perturbed magnetic field
jcr||
-4
-2
0
2
4
-2 0 2 4log10(k)
log1
0(om
ega) Re(ω)
Im(ω) k in units of rg
-1
ω in units of vS2/crg
Magnetic tension inhibits instability Wavelength longer than Larmor radius CR follow field lines. jxB drives weak instability
Bju×−= CRdt
dρ
( )BuB××∇=
∂∂
t
2/10
=
ργ CRjkB
Dispersion relation
Red line is growth rate
k
ω
The essence of the non-resonant instability
j x B j x B
Spiral expands leaving central cavity
Same without vertical field j x B j x B
j x B j x B
Simplest form: expanding loops of B
jxB expands loops
stretches field lines
more B
more jxB
B
CR current
j x B j x B
Simplest form: expanding loops of B
jxB expands loops
stretches field lines
more B
more jxB
B
CR current
Slices through |B| - time sequence (fixed CR current)
Non-linear growth – expanding loops
Cavities and walls in |B| & ρ
Field lines: wandering spirals
How large does the magnetic field grow?
Kepler 1604AD Tycho 1572AD
SN1006 Cas A 1680AD
Chandra observations
NASA/CXC/NCSU/ S.Reynolds et al.
NASA/CXC/Rutgers/ J.Warren & J.Hughes et al.
NASA/CXC/MIT/UMass Amherst/ M.D.Stage et al.
NASA/CXC/Rutgers/ J.Hughes et al.
Historical SNR
Instability growth
a)
d)
b)
No reason for non-linear saturation of a single mode
c)
Saturation (back of envelope)
Magnetic field grows until 1) 2) Set and eliminate between 1) & 2)
( ) BjBB CR ×≈×∇×0
1µ
Magnetic tension CR driving force
LeBp
≈
CR Larmor radius scalelength
L1
=∇
cu
epjB shockCR
3
0
2
efficiency ρµ
×≈≈
L
B2/(8πρ) (cgs)
velocity
Inferred downstream magnetic field (Vink 2008)
Data for RCW86, SN1006, Tycho, Kepler, Cas A, SN1993J
Fit to obs (Vink): Gcmkms10
7002/1
3
2/3
14 µ
≈ −−
enuB
G1.0cmkms10
4002/12/1
3
2/3
14 µη
≈ −−
enuBTheory:
The cosmic ray spectrum
revision from p-4
Shock acceleration energy spectrum: loss rate
High velocity plasma
Low velocity plasma
Upstream ISM Downstream shocked plasma
B2
B1 B2>B1
CR cross from upstream to downstream at rate nshockc/4 CR carried away downstream at rate = ndownstream vs/4 Fraction lost at each shock crossing shock
downstreams
nn
cnn v
−=∆
In diffusive limit shockdownstream nn =
-0.5
0
0.5
1
1.5
u/c=0.1 ν/ωg=0.1
f0
f1
f2
-240 140 x/rg
Parallel shock
Vlasov-Fokker-Planck (VFP) analysis for oblique magnetic field
)(.)v( fCfepfp
xu
xfu
tf
xxx =
∂∂
×+∂∂
∂∂
−∂∂
++∂∂
pBv
Extra term
pppf
pp
pfpppfpff z
zy
yx
x )()()()(0 +++=
Equivalent form of solution for small u/c
Cannot match & across the shock yf zf
Upstream solution
013 f
cufx −= 022
13 fcuf
x
zy ων
νω+
= 022
213 f
cuf
x
zz ων
ω+
=
Downstream solution
0=== zyx fff pme
γBω =
Extra term
pppf
pp
pfpppfpff z
zy
yx
x )()()()(0 +++=
Equivalent form of solution for small u/c
Cannot match & across the shock yf zf
Upstream solution
013 f
cufx −= 022
13 fcuf
x
zy ων
νω+
= 022
213 f
cuf
x
zz ων
ω+
=
Downstream solution
0=== zyx fff pme
γBω =
NOT VALID SOLUTION
)(.)v( fCfepfp
xu
xfu
tf
xxx =
∂∂
×+∂∂
∂∂
−∂∂
++∂∂
pBv
www.trinnov-audio.com/images/sphericalHarm.jpg
( ) ( )∑=ml
ml
ml imPtxftxf
,exp)(cos,,),,( φϑpp
Expand in spherical harmonics
f00
f1m
f2m
f3m
Equation for evolution of each spherical harmonic
Solve numerically
-0.5
0
0.5
1
1.5
Parallel shock
θ = 0o
u/c=0.1 ν/ωg=0.1
f00
f10
f20
-240 140 x/rg
-0.5
0
0.5
1
1.5
θ = 30o
u/c=0.1 ν/ωg=0.1
f00
f10
f20 Im(f11)
-180 100 x/rg
Density spike at shock as seen by Ostrowski MNRAS 249 551 (1991) Ruffalo, ApJ 515 787 (1999) Gieseler, Kirk, Heavens & Achterberg A&A 345 298 (1999)
Oblique shock (nearly parallel)
-0.5
0
0.5
1
1.5θ = 60o
u/c=0.1 ν/ωg=0.1
f00
f10 Re(f11)
Im(f11)
f20
-60 35 x/rg
Re(f11) represents cross-field drift Im(f11) represents drift along oblique field lines
More perpendicular, less parallel
-1
-0.5
0
0.5
1
1.5
2
Peperpendicular shock
θ = 90o
u/c=0.1 ν/ωg=0.1
f00
f10 Re(f11)
Im(f11) f20
-2.8 1.7 x/rg
CR density reduced at shock
0
0.5
1
1.5
2
1
101
-1 0 1 distance from shock (CR Larmor radius)
CR density
θ = 90o
θ = 60o
θ = 72o
upstream downstream
u/c=0.1
ν/ωg=0.1
CR density profiles near shock
shock
3.5
4
4.5
5
5.5
6
6.5
0 1
u=c/10
0 1
Spectral index plotted against shock obliquity
Perpendicular shock
Parallel shock
ν/ωg=0.03
ν/ωg=0.1
ν/ωg=0.3
ν/ωg=1
ν = collision frequency ωg = Larmor frequency
Shock compression x4 in all cases
cosθ
6 (1.5) 5 (1.0) 4 (0.5)
γ (α)
Radio spectral index In brackets
3.5
4
4.5
5
5.5
6
6.5
0 1
3.5
4
4.5
5
5.5
6
6.5
0 1
u=c/30 u=c/10 u=c/5 6 (1.5)
5 (1.0) 4 (0.5)
γ (α)
cosθ 0 1 0 1 0 1
ν/ωg=0.03
ν/ωg=0.1
ν/ωg=0.3 ν/ωg=1
Spectral index plotted against shock obliquity
Perpendicular shock
Parallel shock
ν/ωg=0.03
ν/ωg=0.1
ν/ωg=0.3
ν/ωg=1
Radio spectral index In brackets ν = collision frequency
ωg = Larmor frequency
Shock compression x4 in all cases
cosθ cosθ
ν/ωg=0.03
ν/ωg=0.1
ν/ωg=0.3
ν/ωg=1
Observations
Cosmic Ray spectrum arriving at earth (Nagano & Watson 2000)
Leakage from galaxy accounts for some of difference (Hillas 2005)
E -2
E -2.7
Observed radio spectral index v. mean expansion velocity (following Glushak 1985)
3.5
4
4.5
5
5.5
6
6.5
0 1
Spectral steepening suggests quasi-perpendicular shocks
For random field orientation, steepening & flattening nearly cancel out Steepening at high velocity might be due to 1) expansion into Parker spiral
2) magnetic field amplification jxB stretches field perpendicular to shock normal
u=c/10
0 1
ν/ωg=0.03
ν/ωg=0.1
ν/ωg=0.3
ν/ωg=1
cosθ
6 (1.5) 5 (1.0) 4 (0.5)
γ (α)
SN1006 (Chandra) CR electrons (10-100TeV)
SNR morphology: spectral steepening/flattening
Could be: 1) Quasi-perpendicular shocks accelerate fewer CR to high energy 2) Poor injection at quasi-perpendicular shocks
bi-polar emission (Rothenflug et al 2004)
What we think we know, and what we know we don’t know
What we think we know • Galactic CR are accelerated to 1015
eV by diffusive shock acceleration by SNR
• Streaming CR amplify the magnetic field which confines CR near shock
• CR spectra are often steeper than p -4 (E -2)
What we don’t know • How CR reach 1016 -1017eV
• When CR are accelerated to what energy at different stages of SNR evolution
• How CR escape SNR without losing energy adiabatically
• When & where non-linear effects are important
• Why typically the spectrum is flatter than p -4 in older SNR
• Why is the spectrum so straight?
• Whether second order Fermi acceleration contributes substantially
• Whether perpendicular shocks are good injectors of low energy CR
• How the above applies extra-galactically - the origin of 1020eV CR
